For solving inverse boundary value problems, boundary observation or inside observation have been used. In the present study boundary values on the incompletely prescribed boundary are estimated using the boundary and the inside observation values. The inverse boundary value problem is treated for two-dimensional elastostatic body. Regularization is necessary to obtain a good solution of matrix equation in inverse boundary value problem. We regularize the solution with the use of alternating inverse boundary element method. The number of stopping iteration is necessary for solving inverse boundary value problem with the method. The discrepancy principle that evaluates the discrepancy in observation space or in observation equation was successfully applied for determination of the number of stopping iteration. The effect of the number of stopping iteration on the estimation is discussed.